OK I'm going to describe the purpose of re-amping first:
An amp wants to see a -25 to -20 dBu signal level. That's what comes out of your guitar. A recording device wants to see either a microphone level or line level signal, depending on the device. Most DI boxes put out both signal levels, and are switchable or have separate outputs for each. Mic signal level is about -15 dBu, and line level for pro gear is +4 dBu. (Line level for hobbyist gear is -10 dBv, which is a completely different measurement, note the "v" instead of the "u").
You use the DI box to turn a guitar level signal into a line level signal. You use the Re-amping box to turn the line level signal that comes out of the DAW into a guitar level signal again, so you can amplify it. You can use a real amp to do this with a mic or a load box, or you can use a hardware modeling device to create a line-level signal to plug directly into your line level inputs on your DAW.
There is a third alternative for which you do not need to use a re-amping box: Using the DI to record a dry signal into your DAW, and then choosing a software modeler like the Amp Designer built into Logic to model an amp. Using a software modeler is like using an effects processor in the DAW; it does not change the direct signal permanently (unless you bounce a track).
Since you have ruled out re-amping with a real amp due to noise considerations, you are left with choosing between a hardware modeler using your Re-Amp box, or using software and not needing a Re-amp box.
Also, if you record with a hardware device, you don't need a DI box, all hardware modelers put out a line level signal.
What you do NOT want to do is re-amp the amplified signal you recorded with your amp and the load box. That's an amplified signal that doesn't need re-amping, unless you just want to use it as a wacky effect (which is fine, it's just probably not what you want).
Most DI boxes have an instrument level parallel output that allows you to record both a dry signal into your DAW, and send the other output into your guitar amp or hardware modeler, so you can do both at the same time. This method lets you record both ways at once, so you can have both the amplified signal and the dry signal, and choose between the re-amped signal later, or the wet signal laid down with the amp or modeler, or blend them.
In fact, a good DI box will also have a ground lift to prevent ground loops when doing this.
This is the way it's done in studios.